Lyapunov Exponents and Spectral Properties of Aperiodic Structures

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Order and disorder are familiar concepts to humans. Our brains have evolved to recognise and to appreciate aspects of symmetry and order in nature, as well as in architecture, arts or music. Essentially, science is about detecting and describing ordered patterns in the world around us, and understanding them in mathematical terms. It is thus surprising that it appears to be difficult to give a precise mathematical definition of the concept of order, and indeed there is currently no generally accepted definition available. Consequently, we lack a complete understanding of what types of order are possible, and how to classify them. It turns out to be useful to take a guide from nature. The type of order considered in this project is inspired by physics and crystallography, more precisely the surprising existence of intricately ordered materials called quasicrystals. Their discovery was acknowledged with the award of a Nobel Prize in Chemistry in 2011. As an abstraction of the order of atoms in such materials, mathematicians have investigated the order in patterns or tilings of space.The project will consider a particular type of tilings, which are based on specific rules, in which a tiling can be constructed recursively. Some of these rules lead to particularly nice tilings, which are reasonably well understood. Here we mainly concentrate on tilings that lie outside this class, and investigate their properties, using a novel approach. This promises to provide new insight into order properties of such tilings, which would be a big step towards a classification of order in spatial structures, and will shed new light on the so-called Pisot substitution conjecture, one of the long-standing conjectures in the field that has so far eluded a proof. The new approach also provides a link between two very different characterisations of aperiodic tilings by means of spectral properties. One of these is linked to diffraction as a measure of order, which uses a concept from crystallography and is intimately linked to a mathematical concept of spectrum used in dynamical systems theory. The other spectral characterisation is inspired by studying the physics of electron transport in aperiodic structures, and considers the electronic energy spectrum. These two spectral quantities behave rather differently, and the aim of the project is to understand this and relate these to each other.While the proposed research is fundamental in nature, order phenomena are ubiquitous in nature and an improved understanding of order will be useful in many areas of science. Also, there are many potential applications of aperiodic structures of this type. The most promising are probably in manufactured materials, such as new light-weight strong materials with designed properties that could be used in engineering or medical applications. Aperiodic tilings often are also aesthetically appealing and have increasingly been used in arts and in architecture.